I appreciate any example that explains the procedure in deriving the reaching time for a SMC solution
Hi,
if there exist a Lypunov Function, you can make a simple integration to get the time. you can refer to baht/Bernstein paper 2005.
Geometric homogeneity with applications to finite-time stability.
Regards
I think it's related to the type of sliding surface( linear, or nonlinear) and also type of your function (sgn, sat, sig, AI,.......). As you know SMC has two main parts: corrective control and equivalent control. reaching time is a part of corrective control and used to compensate the deviations from the sliding surface to reach the sliding surface.
Proving the Stability and finite time convergence of a SMC algorithm is really a critical issue.
The reaching time estimation do not have any general description.
The homogeneity proves the finite time convergence effectively but if a perturbed system is assumed then homogeneity becomes a question mark.
The best and hard tool is to find a Lyapunov function for proving the stability and then by the virtue of stability analysis finding the reaching time is abvious.
However keep in mind that the reaching time must depend upon the controller parameters e.g. gain of the controller and the initial conditions.
Moreover if the system is under the effect of some non-vanishing perturbations then the concept of uniform ultimate boundedness changes the definition of reaching time.
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